I normally check the route ahead of time for any possible big rig snafus, but somehow I missed this one. Jackson, WY to West Yellowstone, MT is a commonly driven route, and Google Maps happily takes you up via WY-22, up and over Teton Pass. This pass includes 10% grades for 5 miles on either side, with winding, curving roads. Let me just say that this will be a good test on your engine coolant, engine oil and transmission cooling systems for a motorhome towing a car. We made it fine and I would drive it again. Just take it easy on the way up and keep an eye on all your temperature gauges. Use your exhaust brake and gear down on the opposite side.
An alternative route, which adds about an hour, is to drive through both Grand Teton National Park and Yellowstone National Park via the west side of the Yellowstone Grand loop: follow signs toward Old Faithful over Craig Pass, to Madison Jct and then head west to where park exits into West Yellowstone. It’s a much easier drive (although look out for bison herds and bear jams!) and has gorgeous scenery.